• Corpus ID: 221703717

# Hermitian K-theory for stable $\infty$-categories I: Foundations

@article{Calms2020HermitianKF,
title={Hermitian K-theory for stable \$\infty\$-categories I: Foundations},
author={Baptiste Calm{\e}s and Emanuele Dotto and Yonatan Harpaz and Fabian Hebestreit and Markus Land and Kristian Jonsson Moi and Denis Nardin and Thomas Nickelsen Nikolaus and Wolfgang Steimle},
journal={arXiv: K-Theory and Homology},
year={2020}
}`
• Published 15 September 2020
• Mathematics
• arXiv: K-Theory and Homology
This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $\infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In this article we lay the foundations of our approach by considering Lurie's notion of a Poincare $\infty$-category, which permits an abstract counterpart of unimodular forms…
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## References

SHOWING 1-10 OF 59 REFERENCES
A universal characterization of higher algebraic K-theory
• Mathematics
• 2010
In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is
Stable moduli spaces of hermitian forms
• Mathematics
• 2021
We prove that Grothendieck-Witt spaces of Poincar\'e categories are, in many cases, group completions of certain moduli spaces of hermitian forms. This, in particular, identifies Karoubi's classical
Parametrized higher category theory and higher algebra: Expos\'e IV -- Stability with respect to an orbital $\infty$-category
In this paper we develop a theory of stability for $G$-categories (presheaf of categories on the orbit category of $G$), where $G$ is a finite group. We give a description of Mackey functors as
On the axiomatic foundations of the theory of Hermitian forms
• C. Wall
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 1970
In recent work on some topological problems (7), I was forced to adopt a complicated definition of ‘Hermitian form’ which differed from any in the literature. A recent paper by Tits(5) on quadratic
Topological cyclic homology
• Mathematics
Handbook of Homotopy Theory
• 2020
Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace
On the parametrized Tate construction and two theories of real $p$-cyclotomic spectra
• Mathematics
• 2019
We give a new formula for $p$-typical real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for $p$-typical topological cyclic homology to one
The dualizing spectrum of a topological group
Abstract. To a topological group G, we assign a naive G-spectrum $D_G$, called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that $D_G$ detects Poincaré
Spectral Mackey functors and equivariant algebraic K-Theory ( I )
Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable∞-category, and we use